How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 5?
Volume of largest rectangular box is
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To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane (x + 2y + 3z = 5), follow these steps:

The box will have one vertex on the plane (x + 2y + 3z = 5) and the other three vertices in the coordinate planes. This means that the box will be aligned with the coordinate axes.

Let (x), (y), and (z) represent the lengths of the sides of the rectangular box.

The point ((x, y, z)) that lies on the plane (x + 2y + 3z = 5) must satisfy this equation. Substitute the coordinates of the vertex into the equation to find (z).

Since the other three vertices are on the coordinate planes, their coordinates are ((0, 0, 0)), ((x, 0, 0)), and ((0, y, 0)).

The volume (V) of the rectangular box is given by (V = xyz).

Substitute the values of (x), (y), and (z) obtained in steps 3 and 4 into the formula for volume to find the expression for (V).

To find the maximum volume, differentiate the expression for (V) with respect to (x), (y), and (z), and set the resulting partial derivatives equal to zero.

Solve the resulting system of equations to find the values of (x), (y), and (z).

Substitute these values into the expression for (V) to find the maximum volume of the rectangular box.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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